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- https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Active_Calculus_(Boelkins_et_al.)/11%3A_Multiple_Integrals/11.08%3A_Triple_Integrals_in_Cylindrical_and_Spherical_CoordinatesThe spherical coordinates of a point P in 3-space are ρ (rho), θ, and ϕ (phi), where ρ is the distance from P to the origin, θ is the angle that t...The spherical coordinates of a point P in 3-space are ρ (rho), θ, and ϕ (phi), where ρ is the distance from P to the origin, θ is the angle that the projection of P onto the xy-plane makes with the positive x-axis, and ϕ is the angle between the positive z axis and the vector from the origin to P. When P has Cartesian coordinates (x,y,z), the spherical coordinates are given by
- https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus/15%3A_Multiple_Integration/15.05%3A_Triple_IntegralsIn Double Integrals over Rectangular Regions, we discussed the double integral of a function f(x,y) of two variables over a rectangular region in the plane. In this section we define the triple integ...In Double Integrals over Rectangular Regions, we discussed the double integral of a function f(x,y) of two variables over a rectangular region in the plane. In this section we define the triple integral of a function f(x,y,z) of three variables over a rectangular solid box in space, R³. Later in this section we extend the definition to more general regions in R³.
- https://math.libretexts.org/Courses/Mission_College/Math_4A%3A_Multivariable_Calculus_v2_(Reed)/15%3A_Multiple_Integration/15.04%3A_Triple_IntegralsIn Double Integrals over Rectangular Regions, we discussed the double integral of a function f(x,y) of two variables over a rectangular region in the plane. In this section we define the triple integ...In Double Integrals over Rectangular Regions, we discussed the double integral of a function f(x,y) of two variables over a rectangular region in the plane. In this section we define the triple integral of a function f(x,y,z) of three variables over a rectangular solid box in space, R³. Later in this section we extend the definition to more general regions in R³.
- https://math.libretexts.org/Bookshelves/Calculus/CLP-3_Multivariable_Calculus_(Feldman_Rechnitzer_and_Yeager)/03%3A_Multiple_Integrals/3.05%3A_Triple_IntegralsTriple integrals, that is integrals over three dimensional regions, are just like double integrals, only more so. We decompose the domain of integration into tiny cubes, for example, compute the contr...Triple integrals, that is integrals over three dimensional regions, are just like double integrals, only more so. We decompose the domain of integration into tiny cubes, for example, compute the contribution from each cube and then use integrals to add up all of the different pieces. We'll go through the details now by means of a number of examples.
- https://math.libretexts.org/Courses/University_of_Maryland/MATH_241/04%3A_Multiple_Integration/4.05%3A_Triple_IntegralsIn Double Integrals over Rectangular Regions, we discussed the double integral of a function f(x,y) of two variables over a rectangular region in the plane. In this section we define the triple integ...In Double Integrals over Rectangular Regions, we discussed the double integral of a function f(x,y) of two variables over a rectangular region in the plane. In this section we define the triple integral of a function f(x,y,z) of three variables over a rectangular solid box in space, R³. Later in this section we extend the definition to more general regions in R³.
- https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/13%3A_Multiple_Integration/13.06%3A_Volume_Between_Surfaces_and_Triple_IntegrationThe boundary of D, the line from (0,0,1) to (2,4,0), is shown in part (b) of the figure as a dashed line; it has equation y=2x. (We can recognize this in two ways: one, in collapsing t...The boundary of D, the line from (0,0,1) to (2,4,0), is shown in part (b) of the figure as a dashed line; it has equation y=2x. (We can recognize this in two ways: one, in collapsing the line from (0,0,1) to (2,4,0) onto the x-y plane, we simply ignore the z-values, meaning the line now goes from (0,0) to (2,4).
- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/04%3A_Multiple_Integration/4.04%3A_Triple_IntegralsIn Double Integrals over Rectangular Regions, we discussed the double integral of a function f(x,y) of two variables over a rectangular region in the plane. In this section we define the triple integ...In Double Integrals over Rectangular Regions, we discussed the double integral of a function f(x,y) of two variables over a rectangular region in the plane. In this section we define the triple integral of a function f(x,y,z) of three variables over a rectangular solid box in space, R³. Later in this section we extend the definition to more general regions in R³.
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/Interactive_Calculus_Q4/02%3A_Multiple_Integration/2.05%3A_Triple_IntegralsIn Double Integrals over Rectangular Regions, we discussed the double integral of a function f(x,y) of two variables over a rectangular region in the plane. In this section we define the triple integ...In Double Integrals over Rectangular Regions, we discussed the double integral of a function f(x,y) of two variables over a rectangular region in the plane. In this section we define the triple integral of a function f(x,y,z) of three variables over a rectangular solid box in space, R³. Later in this section we extend the definition to more general regions in R³.
- https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/03%3A_Multiple_Integrals/3.03%3A_Triple_IntegralsWhile the double integral could be thought of as the volume under a two-dimensional surface. It turns out that the triple integral simply generalizes this idea: it can be thought of as representing th...While the double integral could be thought of as the volume under a two-dimensional surface. It turns out that the triple integral simply generalizes this idea: it can be thought of as representing the hypervolume under a three-dimensional hypersurface in R4 . In general, the word “volume” is often used as a general term to signify the same concept for anynn -dimensional object (e.g. length in R1 , area in R2 ).
- https://math.libretexts.org/Courses/Mission_College/Math_4A%3A_Multivariable_Calculus_(Kravets)/04%3A_Multiple_Integration/4.04%3A_Triple_IntegralsIn Double Integrals over Rectangular Regions, we discussed the double integral of a function f(x,y) of two variables over a rectangular region in the plane. In this section we define the triple integ...In Double Integrals over Rectangular Regions, we discussed the double integral of a function f(x,y) of two variables over a rectangular region in the plane. In this section we define the triple integral of a function f(x,y,z) of three variables over a rectangular solid box in space, R³. Later in this section we extend the definition to more general regions in R³.
- https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Vector_Calculus/3%3A_Multiple_Integrals/3.5%3A_Triple_Integrals_in_Rectangular_CoordinatesJust as a single integral has a domain of one-dimension (a line) and a double integral a domain of two-dimension (an area), a triple integral has a domain of three-dimension (a volume). Furthermore, a...Just as a single integral has a domain of one-dimension (a line) and a double integral a domain of two-dimension (an area), a triple integral has a domain of three-dimension (a volume). Furthermore, as a single integral produces a value of 2D and a double integral a value of 3D, a triple integral produces a value of higher dimension beyond 3D, namely 4D.
